What are the critical values, if any, of #f(x)=(4x)/(x^2 -1)#?

Answer 1

#f# has no critical numbers.

#f'(x) = -(4(x^2+1))/(x^2-1)^2#
#f'(x) = 0# never
#f'(x)# does not exist only at #x=+-1# neither of which are in the domain of #f#.
Therefore, #f# has no critical numbers.
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

The critical values of the function ( f(x) = \frac{4x}{x^2 - 1} ) occur where the derivative of the function is equal to zero or undefined. To find the critical values, we first need to find the derivative of the function ( f(x) ), then solve for ( x ) when the derivative is zero or undefined.

Taking the derivative of ( f(x) ) with respect to ( x ) using the quotient rule:

[ f'(x) = \frac{(4)(x^2 - 1) - (4x)(2x)}{(x^2 - 1)^2} ]

Simplifying the expression:

[ f'(x) = \frac{4x^2 - 4 - 8x^2}{(x^2 - 1)^2} ] [ f'(x) = \frac{-4x^2 - 4}{(x^2 - 1)^2} ]

To find critical values, set the derivative equal to zero and solve for ( x ):

[ -4x^2 - 4 = 0 ] [ -4x^2 = 4 ] [ x^2 = -1 ]

This equation has no real solutions, which means there are no critical values for this function.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7